If Hubble's constant is $2.33 \times 10^{-18} \text{ s}^{-1}$ and the earth orbits the sun with average distance of 150 million kilometers; Does that mean the earth's orbital radius increases approximately $11\text{ m}/\text{year}$? Does the earth's angular momentum change? If so, where does the torque come from? If the angular momentum doesn't change, does the earth's orbital velocity (length of a year) change? If so, where does the lost kinetic energy go?

Aside: the 11 meters per year figure comes from Hubble expansion of space the distance of the earth's orbital radius integrated over an entire year.

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No. Hubble's constant roughly says how the distance between two objects at rest with the universe grows. It does not say that the distant between everything is growing - the size of the hydrogen atom is not increasing. (My size is increasing, but from dietary rather than cosmological sources.) The size of objects and orbits are maintained by a balance of forces (classically). To whatever extent one can think of the expansion of the universe as pushing the Earth and Sun apart, it is already taken into account in setting the Earth's orbit.

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The change in the Hubble constant can effect the orbit, see the paper linked by Ben Crowell. But just taking the Hubble constant and multiplying it by the Earth's radius, as I believe you have done, does not give you anything sensible.

looks like you don't believe in the Big Rip. Regardless, there does appear to be some evidence for hubble expansion of the moon's orbit. Though it's a bit more difficult to laser measure the distance from the earth to the sun.
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raeJul 22 '11 at 21:37

Actually a system like the solar system is predicted to expand due to cosmological expansion, but the effect is calculated to be incredibly small, much too small to detect: arxiv.org/abs/astro-ph/9803097v1 The size of a hydrogen atom doesn't change at all, because it's fixed by fundamental constants. Since the solar system does expand by some tiny amount, there is predicted to be a small violation of conservation of energy. That's OK, because energy isn't conserved in general relativity.
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Ben CrowellJul 22 '11 at 21:44

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@rae: The paper by Dumin was never published, and looks just plain wrong to me. It contradicts the Cooperstock paper that I referenced above, which was published in a peer-reviewed journal. There is not a scrap of GR anywhere in the Dumin paper; to my knowledge, no competent relativist has ever suggested that GR leads to an effect of the order of magnitude of the discrepancy that Dumin attributes to cosmological effects. The Big Rip is not really relevant. We don't know if the laws of physics are such as would cause a Big Rip, and the OP is not asking about the remote future.
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Ben CrowellJul 22 '11 at 21:54

@Ben Crowell: I agree that a changing Hubble constant effects the orbit which is what that paper derives (see Eqn. 4.2 depends only on the second derivative of the scale factor). I'm dealing with only the effects of a constant Hubble 'constant', since I believe that what the original poster was asking about. His number 11m/year comes I believe from multiplying the earth's orbital radius by the Hubble constant which is certainly not correct.
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BebopButUnsteadyJul 22 '11 at 22:30

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@rae -- The problem is that there's no clearly-defined meaning to be attached to the phrase "the same point in its orbit the following year." If you use comoving coordinates (i.e., coordinates that expand with the Universe), then "the same point" will be further out than before. If you use local Minkowski coordinates, it won't. And of course there are infinitely many other choices. The common mistake people make is to think that comoving coordinates are what space is "really" doing, but the central idea of relativity is that coordinate systems are just conveniences, not "Truth."
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Ted BunnJul 23 '11 at 17:43

For objects smaller than cosmic scale, such as atoms, planets and solar systems, the electromagnetic and gravitational forces that hold them together are not changing (as far as we know) and so those objects do not change size.

Between galaxies, so widely separated, there's just gravity, and that tends to average out due to every galaxy being surrounded by other galaxies in all directions. On a cosmic scale, galaxies are like a gas, with galaxies being the "molecules" and described by the idea gas equation. To account for gravity and finite size of the galaxies, we might use the Van der Waals equation or some other variation, but that's beside the point, useful only for increasing accuracy.

Hubble's constant describes the rate at which the "container" of the galactic gas is expanding, the way the density of galaxies decreases over time. In an ordinary gas such as air, when in an expanding chamber, certainly the molecules are not expanding. Likewise, neither are the galaxies changing their sizes, at least not for Hubble-related reasons.

You're oversimplifying by treating atoms and solar systems as being the same. GR does predict that solar systems will expand, just not by very much: arxiv.org/abs/astro-ph/9803097v1 The size of a hydrogen atom is set by fundamental constants. The size of a solar system is not.
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Ben CrowellJul 25 '11 at 0:39

The reason the universe expands is gravitation, as described by Einstein's field equation.
The evolution of the universe is governed by gravitation, as described by Einstein's field equation. Over cosmological scale, the universe can be seen as homogeneous and isotropic, with very small density of matter and radiation. The density of matter and radiation is too small to counteract the expansion, an effect of initial condition. In local areas, however, the density is many magnitudes higher, and the effect of expansion is all but counteracted by the binding gravitational attraction.

Everything you've said is true, but it fails to answer the OP's question. One way to see that it doesn't answer it is that although you mention the Einstein field equation, everything you say is equally valid in a Newtonian expanding universe. The fractional rate of change in size due to cosmological expansion is 0 for a hydrogen atom, $\sim 10^{-41}\ \text{s}^{-1}$ for the earth-sun system (arxiv.org/abs/astro-ph/9803097v1 ), and $\sim H_o$ for a photon. I don't see how you would get that from "tends to confine."
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Ben CrowellMar 11 '12 at 1:44

I also wouldn't agree that "the reason the universe expands is gravitation." The reason it has been expanding, ever since the Big Bang, is inertia, and this would be just as true in a Newtonian model as in one based on GR.
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Ben CrowellMar 11 '12 at 1:49

@BenCrowell: To your second comment: By inertial the expansion will slow down, whereas in fact the expansion is speeding up, currently modeled by a non-zero cosmological constant, an effect of gravitation.
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Siyuan RenMar 11 '12 at 1:58

@BenCrowell: To your first comment: I read your paper, and all I see is that according to the authors themselves, this topic, the effect of cosmological expansion on local systems, is highly contentious. I doubt your paper has settled the problem and become the consensus.
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Siyuan RenMar 11 '12 at 2:04

@BenCrowell: Regardless, it is well known that gravitation bounds the solar system. Even cosmological expansion has an effect, it is infinitesimal small. I don't see how that invalidates my phrase "tends to confine" at local scale.
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Siyuan RenMar 11 '12 at 2:08